What's the use of unit vectors?

[Odette]

Homework Statement

Hallo. Can somebody explain to me what's the importance-use of unit vector in the below (second) equation? Why isn't the first equation just enough to describe r? What's the reason for unit vector to even exist?

Relevant Equations

in the photos

Homework Statement: Hallo. Can somebody explain to me what's the importance-use of unit vector in the below (second) equation? Why isn't the first equation just enough to describe r? What's the reason for unit vector to even exist?
Homework Equations: in the photos

 

[PeroK]

$x, y, z$ are numbers. E.g. if $x =1, y =2, z=3$, then $r =6$. Which doesn't mean a lot in terms of 3D vectors.

$\vec{r} = (x, y, z)$ is an alternative that doesn't use unit vectors explicitly, although they are implied in terms an orthonormal basis.

 

[hutchphd]

Suppose I want a constant vector of length a that points in the x directton?
$$\vec{a}=a\hat{x}$$
Or a variable vector of size y that points in z direction? $$\vec{b}=y\hat{z}$$
You really need all the symbols.

 

[callieferg]

Every vector has both a direction and a magnitude. For example, the "x" is the scalar quantity that tells you the magnitude of the x component of r, and the x with the ^ tells you the scalar points in the x direction. So, for a vector r, it has a magnitude of "r" multiplied by the r with the ^ to tell you the direction "r" points in.